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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Quantized symplectic actions and $W$-algebras
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by Ivan Losev PDF
J. Amer. Math. Soc. 23 (2010), 35-59 Request permission

Abstract:

With a nilpotent element in a semisimple Lie algebra $\mathfrak {g}$ one associates a finitely generated associative algebra $\mathcal {W}$ called a $W$-algebra of finite type. This algebra is obtained from the universal enveloping algebra $U(\mathfrak {g})$ by a certain Hamiltonian reduction. We observe that $\mathcal {W}$ is the invariant algebra for an action of a reductive group $G$ with Lie algebra $\mathfrak {g}$ on a quantized symplectic affine variety and use this observation to study $\mathcal {W}$. Our results include an alternative definition of $\mathcal {W}$, a relation between the sets of prime ideals of $\mathcal {W}$ and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of $\mathcal {W}$ in the case of classical $\mathfrak {g}$ and the separation of elements of $\mathcal {W}$ by finite-dimensional representations.
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Additional Information
  • Ivan Losev
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • MR Author ID: 775766
  • Email: ivanlosev@math.mit.edu
  • Received by editor(s): August 17, 2007
  • Published electronically: September 18, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 23 (2010), 35-59
  • MSC (2000): Primary 17B35, 53D55
  • DOI: https://doi.org/10.1090/S0894-0347-09-00648-1
  • MathSciNet review: 2552248